Maximal Function Methods for Sobolev Spaces

Cover
Title page
Preface
Notation
Chapter 1. Maximal Functions
	1.1. Hardy–Littlewood maximal function
	1.2. Hardy–Littlewood–Wiener maximal function theorem
	1.3. Lebesgue differentiation theorem
	1.4. A theorem of Stein
	1.5. Restricted maximal function
	1.6. Riesz potential
	1.7. Fractional maximal function
	1.8. Notes
Chapter 2. Lipschitz and Sobolev Functions
	2.1. Lipschitz functions
	2.2. Sobolev spaces
	2.3. Approximation and calculus in Sobolev spaces
	2.4. Sobolev spaces with zero boundary values
	2.5. Weak convergence and Sobolev spaces
	2.6. Difference quotients
	2.7. Notes
Chapter 3. Sobolev and Poincaré Inequalities
	3.1. Pointwise estimates for Lipschitz functions
	3.2. Sobolev–Gagliardo–Nirenberg inequality
	3.3. Sobolev–Poincaré inequalities
	3.4. Poincaré inequalities for zero boundary values
	3.5. Morrey’s inequality
	3.6. Notes
Chapter 4. Pointwise Inequalities for Sobolev Functions
	4.1. Pointwise characterization of Sobolev spaces
	4.2. Lipschitz truncation of Sobolev functions
	4.3. Campanato and Morrey approaches to Sobolev spaces
	4.4. Maximal operator on Sobolev spaces
	4.5. Maximal function with respect to an open set
	4.6. Fractional maximal operator on Sobolev spaces
	4.7. Notes
Chapter 5. Capacities and Fine Properties of Sobolev Functions
	5.1. Sobolev capacity
	5.2. Estimates for capacity
	5.3. Quasicontinuity and fine properties of capacity
	5.4. Lebesgue points of Sobolev functions
	5.5. Sobolev spaces with zero boundary values revisited
	5.6. Variational capacity
	5.7. Capacity and Hausdorff content
	5.8. Lipschitz test functions for variational capacity
	5.9. Maz′ya’s inequality
	5.10. Notes
Chapter 6. Hardy’s Inequalities
	6.1. Introduction to Hardy’s inequalities
	6.2. Measure density and Hardy’s inequality
	6.3. Self-improvement of Hardy’s inequality
	6.4. Capacity density and pointwise Hardy inequalities
	6.5. Wannebo’s approach
	6.6. Stability of Sobolev spaces with zero boundary values
	6.7. Notes
Chapter 7. Density Conditions
	7.1. Hausdorff content density
	7.2. Ahlfors–David regular sets
	7.3. Lower dimension and capacity density
	7.4. Density conditions and Hardy’s inequality in the borderline case
	7.5. Self-improvement of the capacity density condition
	7.6. Truncation and absorption
	7.7. Local Hardy inequality
	7.8. Concluding argument
	7.9. Notes
Chapter 8. Muckenhoupt Weights
	8.1. Doubling weights
	8.2. Dyadic cubes and the Calderón–Zygmund lemma
	8.3. Self-improvement of weighted norm inequalities
	8.4. Muckenhoupt